Thanks for reading me. I have a very particular problem. I want to know whether the following is satisfied:
$\lim_{k \rightarrow \infty} |x_k| = \lim_{k \rightarrow \infty} | (\sum_{j=0}^{k} (A + B)^j) B (\sum_{i=0}^{k}A^i)Z| = \infty$,
where $A,B \in \mathbb{R}^{n \times n}$, $Z \in \mathbb{R}^{n}$, $|\cdot|$ denotes Euclidean norm, $\rho(A) \geq 1$, $\rho(A+B) < 1$ (spectral radius), rank$(B)=m<n$. Any hint would be highly appreciated.